Polarized products of elliptic curves with complex multiplication and field of moduli $\mathbb{Q}$

Abstract

Let $R$ be the maximal order in a quadratic imaginary field $K$. We give an equivalence of categories between the category of polarized abelian varieties isomorphic to a product of elliptic curves over $\mathbb{C}$ with complex multiplication (CM) by $R$ and the category of integral hermitian $R$-lattices. Then we apply this equivalence to enumerate all the genus $2$ and $3$ curves with field of moduli $\mathbb{Q}$ and with Jacobian isomorphic to a product of elliptic curves with CM by $R$.